Abstract

A novel interferometer based on a conventional phase-shifting design is presented. This interferometer is capable of measuring both the real and the imaginary parts of the complex index of refraction and the surface profile of a test surface, n, k, and h, respectively. Maximum-likelihood-estimation theory is shown to be a viable means of extracting the three parameters of interest from the measured data. A Monte Carlo simulation showed limited success in estimating the complex index parameters. The results exhibited bias or deviation from the true values for the system configuration examined. The estimate on the surface profile showed excellent agreement with the true value, although the error in the estimate was an order of magnitude worse than in the case in which only the surface profile is to be estimated.

© 1998 Optical Society of America

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References

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  1. J. E. Greivenkamp, J. H. Bruning, “Phase-Shifting Interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.
  2. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Vol. XXVI, pp. 349–393.
  3. Y. Liu, C. W. See, M. G. Somekh, “Common path interferometric microellipsometry,” in Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 635–645 (1996).
    [CrossRef]
  4. S. V. Shatalin, R. Juskaitis, J. B. Tan, T. Wilson, “Reflection conoscopy and microellipsometry of isotropic thin-film structures,” J. Microsc. 179, pt. 3, 241–252 (1995).
    [CrossRef]
  5. A. Rosencwaig, J. Opsal, D. L. Willenborg, S. M. Kelso, J. T. Fanton, “Beam profile reflectometry: a new technique for dielectric film measurements,” Appl. Phys. Lett. 60, 11,1301–1303 (1992).
    [CrossRef]
  6. N. Gold, D. Willenborg, J. Opsal, A. Rosencwaig, “Method and apparatus for measuring thickness of thin films,” U.S. Patent No.4,999,014 (1991).
  7. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  8. M. Pluta, Advanced Light Microscopy (Elsevier, New York, 1989).
  9. H. L. Van Trees, Detection, Estimation, and Linear Modulation Theory, Part 1 (Wiley, New York, 1968).
  10. M. G. Kendall, A. Stuart, The Advanced Theory of Statistics, Vol. 2, 3rd ed. (Hafner, New York, 1973).
  11. B. R. Frieden, Probability, Statistical Optics, and Data Testing, 2nd ed. (Springer-Verlag, New York, 1991).
  12. D. L. Cohn, J. L. Melsa, Decision and Estimation Theory (McGraw-Hill, New York, 1978).
  13. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 394–455.
  14. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 274–328.
  15. S. K. Park, K. W. Miller, “Random number generators: good ones are hard to find,” Commun. ACM 31, 1192–1201 (1988).
    [CrossRef]
  16. P. L’Ecuyer, “Efficient and portable combined random number generators,” Commun. ACM 31, 742–749 (1988).
    [CrossRef]
  17. B. R. Freiden, Probability, Statistical Optics, and Data Testing, 2nd ed. (Springer-Verlag, New York, 1991), p. 165.
  18. M. Quenouille, “Approximate tests of correlation in time series,” J. R. Statist. Soc. Ser. B, 11, 18–84 (1949).
  19. J. Tukey, “Bias and confidence in not quite large samples,” Ann. Math. Statist. Soc. Ser. B, 29, 614 (1958).
  20. B. Efron, The Jackknife, the Bootstrap and Other Resampling Plans (Society for Industrial and Applied Mathematics, Philadelphia, 1982).
  21. K. M. Wolter, Introduction to Variance Estimation (Springer-Verlag, New York, 1985), pp. 153–200.
  22. F. Mostellar, J. W. Tukey, Data Analysis and Regression: A Second Course in Statistics (Addison-Wesley, Reading, Mass., 1977), pp. 119–162.
  23. E. W. Rogala, H. H. Barrett, “Maximum-likelihood estimation theory and phase-shifting interferometry,” Appl. Opt. 36, 8871–8876.

1995

S. V. Shatalin, R. Juskaitis, J. B. Tan, T. Wilson, “Reflection conoscopy and microellipsometry of isotropic thin-film structures,” J. Microsc. 179, pt. 3, 241–252 (1995).
[CrossRef]

1992

A. Rosencwaig, J. Opsal, D. L. Willenborg, S. M. Kelso, J. T. Fanton, “Beam profile reflectometry: a new technique for dielectric film measurements,” Appl. Phys. Lett. 60, 11,1301–1303 (1992).
[CrossRef]

1988

S. K. Park, K. W. Miller, “Random number generators: good ones are hard to find,” Commun. ACM 31, 1192–1201 (1988).
[CrossRef]

P. L’Ecuyer, “Efficient and portable combined random number generators,” Commun. ACM 31, 742–749 (1988).
[CrossRef]

1958

J. Tukey, “Bias and confidence in not quite large samples,” Ann. Math. Statist. Soc. Ser. B, 29, 614 (1958).

1949

M. Quenouille, “Approximate tests of correlation in time series,” J. R. Statist. Soc. Ser. B, 11, 18–84 (1949).

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Barrett, H. H.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bruning, J. H.

J. E. Greivenkamp, J. H. Bruning, “Phase-Shifting Interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.

Cohn, D. L.

D. L. Cohn, J. L. Melsa, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Vol. XXVI, pp. 349–393.

Efron, B.

B. Efron, The Jackknife, the Bootstrap and Other Resampling Plans (Society for Industrial and Applied Mathematics, Philadelphia, 1982).

Fanton, J. T.

A. Rosencwaig, J. Opsal, D. L. Willenborg, S. M. Kelso, J. T. Fanton, “Beam profile reflectometry: a new technique for dielectric film measurements,” Appl. Phys. Lett. 60, 11,1301–1303 (1992).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 394–455.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 274–328.

Freiden, B. R.

B. R. Freiden, Probability, Statistical Optics, and Data Testing, 2nd ed. (Springer-Verlag, New York, 1991), p. 165.

Frieden, B. R.

B. R. Frieden, Probability, Statistical Optics, and Data Testing, 2nd ed. (Springer-Verlag, New York, 1991).

Gold, N.

N. Gold, D. Willenborg, J. Opsal, A. Rosencwaig, “Method and apparatus for measuring thickness of thin films,” U.S. Patent No.4,999,014 (1991).

Greivenkamp, J. E.

J. E. Greivenkamp, J. H. Bruning, “Phase-Shifting Interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.

Juskaitis, R.

S. V. Shatalin, R. Juskaitis, J. B. Tan, T. Wilson, “Reflection conoscopy and microellipsometry of isotropic thin-film structures,” J. Microsc. 179, pt. 3, 241–252 (1995).
[CrossRef]

Kelso, S. M.

A. Rosencwaig, J. Opsal, D. L. Willenborg, S. M. Kelso, J. T. Fanton, “Beam profile reflectometry: a new technique for dielectric film measurements,” Appl. Phys. Lett. 60, 11,1301–1303 (1992).
[CrossRef]

Kendall, M. G.

M. G. Kendall, A. Stuart, The Advanced Theory of Statistics, Vol. 2, 3rd ed. (Hafner, New York, 1973).

L’Ecuyer, P.

P. L’Ecuyer, “Efficient and portable combined random number generators,” Commun. ACM 31, 742–749 (1988).
[CrossRef]

Liu, Y.

Y. Liu, C. W. See, M. G. Somekh, “Common path interferometric microellipsometry,” in Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 635–645 (1996).
[CrossRef]

Melsa, J. L.

D. L. Cohn, J. L. Melsa, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

Miller, K. W.

S. K. Park, K. W. Miller, “Random number generators: good ones are hard to find,” Commun. ACM 31, 1192–1201 (1988).
[CrossRef]

Mostellar, F.

F. Mostellar, J. W. Tukey, Data Analysis and Regression: A Second Course in Statistics (Addison-Wesley, Reading, Mass., 1977), pp. 119–162.

Opsal, J.

A. Rosencwaig, J. Opsal, D. L. Willenborg, S. M. Kelso, J. T. Fanton, “Beam profile reflectometry: a new technique for dielectric film measurements,” Appl. Phys. Lett. 60, 11,1301–1303 (1992).
[CrossRef]

N. Gold, D. Willenborg, J. Opsal, A. Rosencwaig, “Method and apparatus for measuring thickness of thin films,” U.S. Patent No.4,999,014 (1991).

Park, S. K.

S. K. Park, K. W. Miller, “Random number generators: good ones are hard to find,” Commun. ACM 31, 1192–1201 (1988).
[CrossRef]

Pluta, M.

M. Pluta, Advanced Light Microscopy (Elsevier, New York, 1989).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 394–455.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 274–328.

Quenouille, M.

M. Quenouille, “Approximate tests of correlation in time series,” J. R. Statist. Soc. Ser. B, 11, 18–84 (1949).

Rogala, E. W.

Rosencwaig, A.

A. Rosencwaig, J. Opsal, D. L. Willenborg, S. M. Kelso, J. T. Fanton, “Beam profile reflectometry: a new technique for dielectric film measurements,” Appl. Phys. Lett. 60, 11,1301–1303 (1992).
[CrossRef]

N. Gold, D. Willenborg, J. Opsal, A. Rosencwaig, “Method and apparatus for measuring thickness of thin films,” U.S. Patent No.4,999,014 (1991).

See, C. W.

Y. Liu, C. W. See, M. G. Somekh, “Common path interferometric microellipsometry,” in Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 635–645 (1996).
[CrossRef]

Shatalin, S. V.

S. V. Shatalin, R. Juskaitis, J. B. Tan, T. Wilson, “Reflection conoscopy and microellipsometry of isotropic thin-film structures,” J. Microsc. 179, pt. 3, 241–252 (1995).
[CrossRef]

Somekh, M. G.

Y. Liu, C. W. See, M. G. Somekh, “Common path interferometric microellipsometry,” in Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 635–645 (1996).
[CrossRef]

Stuart, A.

M. G. Kendall, A. Stuart, The Advanced Theory of Statistics, Vol. 2, 3rd ed. (Hafner, New York, 1973).

Tan, J. B.

S. V. Shatalin, R. Juskaitis, J. B. Tan, T. Wilson, “Reflection conoscopy and microellipsometry of isotropic thin-film structures,” J. Microsc. 179, pt. 3, 241–252 (1995).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 394–455.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 274–328.

Tukey, J.

J. Tukey, “Bias and confidence in not quite large samples,” Ann. Math. Statist. Soc. Ser. B, 29, 614 (1958).

Tukey, J. W.

F. Mostellar, J. W. Tukey, Data Analysis and Regression: A Second Course in Statistics (Addison-Wesley, Reading, Mass., 1977), pp. 119–162.

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation, and Linear Modulation Theory, Part 1 (Wiley, New York, 1968).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 394–455.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 274–328.

Willenborg, D.

N. Gold, D. Willenborg, J. Opsal, A. Rosencwaig, “Method and apparatus for measuring thickness of thin films,” U.S. Patent No.4,999,014 (1991).

Willenborg, D. L.

A. Rosencwaig, J. Opsal, D. L. Willenborg, S. M. Kelso, J. T. Fanton, “Beam profile reflectometry: a new technique for dielectric film measurements,” Appl. Phys. Lett. 60, 11,1301–1303 (1992).
[CrossRef]

Wilson, T.

S. V. Shatalin, R. Juskaitis, J. B. Tan, T. Wilson, “Reflection conoscopy and microellipsometry of isotropic thin-film structures,” J. Microsc. 179, pt. 3, 241–252 (1995).
[CrossRef]

Wolter, K. M.

K. M. Wolter, Introduction to Variance Estimation (Springer-Verlag, New York, 1985), pp. 153–200.

Ann. Math. Statist. Soc. Ser. B

J. Tukey, “Bias and confidence in not quite large samples,” Ann. Math. Statist. Soc. Ser. B, 29, 614 (1958).

Appl. Opt.

Appl. Phys. Lett.

A. Rosencwaig, J. Opsal, D. L. Willenborg, S. M. Kelso, J. T. Fanton, “Beam profile reflectometry: a new technique for dielectric film measurements,” Appl. Phys. Lett. 60, 11,1301–1303 (1992).
[CrossRef]

Commun. ACM

S. K. Park, K. W. Miller, “Random number generators: good ones are hard to find,” Commun. ACM 31, 1192–1201 (1988).
[CrossRef]

P. L’Ecuyer, “Efficient and portable combined random number generators,” Commun. ACM 31, 742–749 (1988).
[CrossRef]

J. Microsc.

S. V. Shatalin, R. Juskaitis, J. B. Tan, T. Wilson, “Reflection conoscopy and microellipsometry of isotropic thin-film structures,” J. Microsc. 179, pt. 3, 241–252 (1995).
[CrossRef]

J. R. Statist. Soc. Ser. B

M. Quenouille, “Approximate tests of correlation in time series,” J. R. Statist. Soc. Ser. B, 11, 18–84 (1949).

Other

B. R. Freiden, Probability, Statistical Optics, and Data Testing, 2nd ed. (Springer-Verlag, New York, 1991), p. 165.

J. E. Greivenkamp, J. H. Bruning, “Phase-Shifting Interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Vol. XXVI, pp. 349–393.

Y. Liu, C. W. See, M. G. Somekh, “Common path interferometric microellipsometry,” in Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 635–645 (1996).
[CrossRef]

B. Efron, The Jackknife, the Bootstrap and Other Resampling Plans (Society for Industrial and Applied Mathematics, Philadelphia, 1982).

K. M. Wolter, Introduction to Variance Estimation (Springer-Verlag, New York, 1985), pp. 153–200.

F. Mostellar, J. W. Tukey, Data Analysis and Regression: A Second Course in Statistics (Addison-Wesley, Reading, Mass., 1977), pp. 119–162.

N. Gold, D. Willenborg, J. Opsal, A. Rosencwaig, “Method and apparatus for measuring thickness of thin films,” U.S. Patent No.4,999,014 (1991).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

M. Pluta, Advanced Light Microscopy (Elsevier, New York, 1989).

H. L. Van Trees, Detection, Estimation, and Linear Modulation Theory, Part 1 (Wiley, New York, 1968).

M. G. Kendall, A. Stuart, The Advanced Theory of Statistics, Vol. 2, 3rd ed. (Hafner, New York, 1973).

B. R. Frieden, Probability, Statistical Optics, and Data Testing, 2nd ed. (Springer-Verlag, New York, 1991).

D. L. Cohn, J. L. Melsa, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 394–455.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 274–328.

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Figures (9)

Fig. 1
Fig. 1

Twyman–Green interferometer configuration under consideration. Only a single point is under consideration here.

Fig. 2
Fig. 2

Polarization pupil mask. Transmitting regions defined by rmin, rmax, and α0.

Fig. 3
Fig. 3

Unfolded interferometer arm.

Fig. 4
Fig. 4

Relationship between pupil coordinate and angle of incidence onto the test surface.

Fig. 5
Fig. 5

Surface plots for one realization of data, showing the behavior of the function as a function of two of the parameters of interest, n and k, while the third, h, is held constant. The values of h shown are (a) -0.5 nm, (b) 0.0 nm, and (c) +0.5 nm.

Fig. 6
Fig. 6

Minimization function shown as a function of h for one realization of data; where minimization with respect to n and k has been performed at each value of h.

Fig. 7
Fig. 7

A plot of n and k minimums found for each value of h for one realization of data. The asterisk denotes the parameter values at the minimization function’s bracketed minimum.

Fig. 8
Fig. 8

Scatterplot of the 20,000 ML estimates for the ran2(-1) Monte Carlo experiment.

Fig. 9
Fig. 9

Histogram showing the distribution of the 20,000 ML estimates. (a) on nˆ, (b) on kˆ, (c) on hˆ.

Tables (1)

Tables Icon

Table 1 Monte Carlo Experimental Results for Three Random-Number Generators

Equations (47)

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EV+(x, y)=-E0 exp(i4κf0)tI(x, y)tV(-x,-y)ρ˜(x, y),
ϕ=tan-1x2+y2f0=tan-1rf0.
EV+(x, y)=-E0 exp(i4κf0)ρ˜(x, y).
Efinal(0, 0)=i exp(iκf)exp(iκd)exp(i4κf0)E0λf×pupilρ˜(r)rdrdα,
Eref(0, 0)=i exp(imπ/2)E0refλf02π0Rrdrdα=i exp(imπ/2)λfE0refApupil,
Etest(0, 0)=2 E0test i exp[i(4πh)/λ]λf×-α0α0rminrmaxrρ˜j(r)drdα=4α0E0test i exp[i(4πh)/λ]λfrminrmaxrρ˜j(r)dr.
ρ˜TE(ϕ)=cos(ϕ)-(n+ik)cos(ϕ)cos(ϕ)+(n+ik)cos(ϕ),
ρ˜TM(ϕ)=(n+ik)cos(ϕ)-cos(ϕ)(n+ik)cos(ϕ)+cos(ϕ),
cos(ϕ)=+1-sin(ϕ)n+ik2.
ρ˜TE(r)=cos[tan-1(r/f)]-(n+ik)1-sin[tan-1(r/f)](n+ik)2cos[tan-1(r/f)]+(n+ik)1-sin[tan-1(r/f)](n+ik)2,
ρ˜TM(r)=(n+ik)cos[tan-1(r/f)]-1-sin[tan-1(r/f)](n+ik)2(n+ik)cos[tan-1(r/f)]+1-sin[tan-1(r/f)](n+ik)2.
Ijm(n, k, h)=i exp(imπ/2)λfE0refApupil+4α0E0test i exp[(i4πh)/λ]λfrminrmaxrρ˜j(r)dr2.
rminrmaxρ˜j(r)rdrdα=ξj exp(iγj).
Ijm(n, k, h)=|exp(imπ/2)Eref+exp[(i4πh)/λ]exp(iγj)Etest|2,
Eref=iE0refApupilλf,Etest=4iα0E0testλfξj
Ijm(n, k, h)=Iref+Itest+2IrefItest cosmπ2-4πλh-γj,
Ijm=I¯jm(n, k, h)+ηjm,
pN(ηjm)=12πσNexp-ηjm22σN2.
p(Ijm|θ=n, k, h)=12πσNexp-[Ijm-I¯jm(n, k, h)]22σN2.
p(I|θ)=12πσNj=01m=03 exp-[Ijm-I¯jm(θ)]22σN2.
ln[p(I|θ)]=j=01m=0312πσN-12σN2×[Ijm-I¯jm(θ)]2.
n{ln[p(I|θ)]}θ=θˆML(I)=1σN2j=01m=03{[Ijm-I¯jm(θ)]×n[I¯jm(θ)]}θ=θˆML(I)=0,
k{ln[p(I|θ)]}θ=θˆML(I)=1σN2j=01m=03[Ijm-I¯jm(θ)]×k[I¯jm(θ)]θ=θˆML(I)=0,
h{ln[p(I|θ)]}θ=θˆML(I)=1σN2j=11m=03[Ijm-I¯jm(θ)]×h[I¯jm(θ)]θ=θˆML(I)=0.
Φ(n, k, h)=-p(I|θ)=-12πσN×exp-j=01m=03[Ijm-I¯jm(n, k, h)]22σN2.
0 0=Φn(n(l), k(l), h)Φk(n(l), k(l), h)+2Φn2(n(l), k(l), h)2Φnk(n(l), k(l), h)2Φkn(n(l), k(l), h)2Φk2(n(l), k(l), h)×Δn Δk,
Δn Δk=-2Φn2(n(l), k(l), h)2Φnk(n(l), k(l), h)2Φkn(n(l), k(l), h)2Φk2(n(l), k(l), h)-1×Φn(n(l), k(l), h)Φk(n(l), k(l), h).
n(l+1)=n(l)+Δn,k(l+1)=k(l)+Δk.
Φωi(n(l+1), k(l+1), h)-Φθi(n(l), k(l), h)1e-10,
i=1n,i=2k.
(n(l+1), k(l+1), h(a)+Δh)=(n(1), k(1), h(b)).
Δn Δk Δh=-2Φn22Φnk2Φnh2Φkn2Φk22Φkh2Φhn2Φhk2Φh2-1ΦnΦkΦh.
Φθi(n(l+1), k(l+1), h)-Φθi(nl, k(l), h)1e-10,
i=1n,i=2k,i=3h.
Φ(n, k, h)=-p(I|θ)=-12πσN×exp-j=01m=03[Ijm-I¯jm(n, k, h)]22σN2.
ϕθi=-12πσNexp-j=01m=03[Ijm-I¯jm(n, k, h)]22σN2×j=01m=031σN2[Ijm-I¯jm(n, k, h)]×I¯jm(n, k, h)θi.
Ijm(n, k, h)θi=Em(ref)+Ej(test)θi×Em(ref)+Ej(test)θi*.
Ej(test)h=i4πλEj(test),
Ej(test)n=4α0E0test exp[(i4πh)/λ]iλfrminrmaxr ρ˜j(r)dndr,
Ej(test)k=4α0E0test exp[(i4πh)/λ]iλfrminrmaxr ρ˜j(r)kdr.
σN=18j=01m=03I¯jm.
x=x+σx-2 ln(y1) cos(2πy2).
ξˆi=θˆi,ζˆi=σθˆi2,
i=1n,i=2k,i=3h.
ξˆiγ=Nξˆi-(N-1)ξˆi(γ).
ξˆiJK=γ=1N ξˆiγN,
σξˆiJK2=1N(N-1)γ=1N(ξˆiγ-ξˆi)2.

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